Need for Matching — Impedance Matching

Along the way

1
The source has an \(R_S\). A Thevenin port is \(V_S\) behind \(R_S\). You cannot get around that resistance — it burns power whenever current flows.
2
Voltage and power disagree. \(V_L\) grows with \(R_L\) (divider). Open-circuit voltage is huge; delivered power is zero. Power is \(\tfrac{1}{2}|V_L|^2/R_L\).
3
The peak sits at \(R_L = R_S\). Green dashed line and band: that’s the match. Blue near the peak, red when you leave most of \(P_{\mathrm{avail}}\) on the table.
4
Even at the match, \(R_S\) and \(R_L\) share equally. Each gets \(P_{\mathrm{avail}}\); the ideal source puts out \(2P_{\mathrm{avail}}\). Available power is the load’s ceiling.
5
50 Ω is a convention, not magic. Change \(R_S\) and the peak moves. Cables and instruments picked a standard so “matched” has one number.

Resistive ports only. Real loads are \(Z = R + jX\); the full rule is the conjugate match \(Z_L = Z_S^*\). Parts 2–5 build that language.

Next: Part 2 — \(Z = R + jX\) and how frequency moves \(X\).

Power into \(R_L\)

Fixed source \(R_S\) and open-circuit \(V_S\) · vary the load · log \(R_L\)

\(R_S\) (Ω) \(R_L\) (Ω) \(V_S\) (V peak)

Source V_S behind R_S (blue) into load R_L (rose, to ground). Operating point is V_L.

\(P_L\) vs \(R_L\) · green band ≈ 90% of available · red dot = this load

Status—

V_load—

P_load—

P in R_S—

P_available—

Fraction delivered—

Mismatch loss—

\[ V_L = V_S\,\dfrac{R_L}{R_S + R_L} \]

= —

\[ P_L = \dfrac{1}{2}\,\dfrac{|V_L|^2}{R_L} \]

= —

\[ P_{\mathrm{avail}} = \dfrac{V_S^2}{8\,R_S} \]

= —

Peak source \(V_S\); average powers. Curve is blue near the match and red when \(P_L < 90\%\) of available.

Try it: “Matched” sits on the peak. Switch to “Big \(V_L\)” — voltage is larger, power is worse. Then drag \(R_S\) and watch the green line (and the peak) move with it.